Where is quantum theory headed?

0.09 MB·

by  Stephen L. Adler

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Where is quantum theory headed?
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n Stephen L. Adler
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Institutefor AdvancedStudy,Einstein Drive, Princeton, NJ 08540, USA
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5 E-mail: [email protected]

h] Abstract. PublictalkattheEmQM13conferenceopeningeventon“TheFutureofQuantum
p Mechanics”
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u Theorganizers have asked me tostate my views on the direction of thefuturedevelopment of
q quantum mechanics. Will it evolve within the standard framework, without the addition of new
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foundational physics? Or will the foundations require modification in an, at least in principle,

1 experimentally detectable way?
v First, let’s discuss the current status of quantum theory. Quantum mechanics is our most
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successful physical theory. It underlies our detailed understanding of atomic physics, chemistry,
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8 and nuclear physics, and the many technologies based on this knowledge. Additionally,
0 relativistic quantum mechanics is the basis for the very successful standard model of elementary
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1 particles.
0 However, fromitsbeginningstherehavebeenconceptualproblemsassociated withthenature
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of measurement in quantum mechanics. These can be simply illustrated with the famous Stern-
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: Gerlach experiment (Fig. 1).
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Silver atoms boiled off from a furnace are sent through a non-uniform magnetic field, and
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impinge on a photographic plate. Instead of a continuous distribution of spots, one sees two
r spots, corresponding to spin up and spin down relative to the magnetic field axis. Each atom
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STERN−GERLACH EXPERIMENT
UP
OR
OVEN
MAGNET DOWN
PLATE
Figure 1. Schematic representation of the Stern–Gerlach experiment
goes up OR down, butone cannot predictwhich in any given run–the results of the experiment
are probabilistic. There is a 50% chance of an atom going up, and a 50% chance that it will go
down.
From the point of view of the Schro¨dinger equation of quantum theory, this result has no
explanation. In quantum theory, the state of the particle is described by its wave function, and
the Schro¨dinger equation says that at a post-measurement final time T , the wave function is
f
related to that at a pre-measurement initial time T , by a deterministic relation
i
Ψ(T ) = U(T ,T )Ψ(T ) ,
f f i i
U(Tf,Ti) = eiH(Tf−Ti) ,
with the transition operator U completely specified by the Hamiltonian H. To explain what is
observed, the Schro¨dinger equation must be supplemented by the reduction postulate and the
Born rule. These state that the wave function only gives a description of probabilities when a
measurement is made, with the probabilities for an “up” outcome and a “down” outcome given
by the squares of the coefficients of the corresponding components in the initial wave function
Ψ(T ),
i
Born Rule for Probabilities
Ψ(T ) = C Ψ +C Ψ ,
i up up down down
prob = |C |2 ,
up up
prob = |C |2 ,
down down
|C |2+|C |2 = 1 ,
up down
with the sum of the up and down probabilities equal to one. The reduction postulate and Born
rule are an add-on to the Schro¨dinger equation. According to the Copenhagen interpretation
of quantum mechanics, the Schro¨dinger equation applies when a microscopic system, the silver
atom, is time-evolving in isolation. But when the atom interacts with a macroscopic measuring
apparatus, as in the Stern–Gerlach setup, you have to use the reduction postulate and Born
rule.
This situation leads to puzzles that have been debated for over eighty years. If quantum
mechanics describes the whole universe, then why can’t one use the Schro¨dinger equation to
describe the system consisting of the silver atom plus the measuring apparatus? But we never
see a superposition state of the atom plus apparatus. This is Schro¨dinger’s famous cat paradox.
Arrange the experiment so that an“up” outcome triggers a mechanism that kills the cat, while
a “down” outcome keeps the cat alive. Of course we don’t do this, but if we were to do it, we
would always see a live cat OR a dead one, never a superposition of the two (Fig. 2). So we
have the problem of definite outcomes: where does the “either”–“or”dichotomy arise?
A related question is where do the probabilities come from? Quantum mechanics has
probabilities without a sample space! An example of a sample space is a population of people,
40% blonde and 60% brunette. If you pick a person at random from the population, there is
a probability prob = 0.4 that you will have a blonde, and prob = 0.6 that you will
blonde brunette
haveabrunette. Butthepopulation(thesamplespace) iscomposedof individuals,withdefinite
hair coloring – the probabilities only reflect our ignorance of details if we make a random pick
without looking. Another example of a sample space, closer to our Stern–Gerlach experiment,
is a coin toss. Consider 1000 coin tosses. If the coin is tossed without bias, you will find close
to 500 heads and 500 tails, corresponding to prob = 0.5 and prob = 0.5. Here the
heads tails
sample space consists of the 1000 detailed trajectories of the toss, which your eye cannot follow,
SCHRODINGER CAT
UP
KILLS CAT
CAT OK
DOWN
Figure 2. The Stern–Gerlach experiment with a Schro¨dinger cat as the outcome registration
but which if analyzed by a very fast computer could predict which toss would give a head and
which a tail (Fig. 3). Again, the probabilities are just reflections of our ignorance of the details,
but the details are there. So we have the questions – are there hidden details underlying the
probabilities in quantum mechanics? Is there a hidden sample space?
Now we come to the question with which I began – where is future work to deal with these
problems headed? Two routes proceed within quantum theory: (i) The first route within
quantum theory is to try to change the interpretational rules. Examples are the so-called
“many worlds” interpretation (all possibilities are there, we just only see one), and the so-called
“histories” program, which sets up an observer-free generalization of the Copenhagen rules. (ii)
The second route within quantum theory is to say there is a sample space, but we don’t see
SAMPLE SPACE
(a)
BLONDE BRUNETTE
p = 0.4 p = 0.6
(b)
DETAILED TRAJECTORY
COIN
HEADS p = 0.5
OR TAILS p = 0.5
Figure 3. Some sample spaces: (a) Populations of individuals with blonde/brunette hair
coloring (b) Trajectories in a coin toss
it. One example is the proposal of Bohmian trajectories as the sample space underlying the
Born rule. Another example comprises various statistical interpretations, which attribute the
probabilistic outcomes to different internal states of the apparatus and or its environment, with
the aim of deriving the Born rule.
None of these routes within orthodox quantum theory has gained general acceptance. Also,
none makes experimental predictions at odds with standard theory, so experimentally they are
not distinguishable.
The other possibility is to modify the foundations of quantum theory. Specifically, one gets
a sample space by postulating additional degrees of freedom – so called “hidden variables”.
Therearetwopossibilitiesforhiddenvariables–theycanbelocal, ornonlocal. Localvariables
have the property that variables V(x ,t ) and W(x ,t ), with x ,x the spatial points and t ,t
1 1 2 2 1 2 1 2
the times of occurrence, cannot influence one another if the distance between them |x −x | is
1 2
greater than thedistance c|t −t |that light can travel, at velocity c, in thetime interval from t
1 2 1
to t . Such local variables are called causally separated. In quantum theory, causally separated
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variables have a commutative multiplication law
V(x ,t )×W(x ,t ) = W(x ,t )×V(x ,t ) .
1 1 2 2 2 2 1 1
Ordinary numbers obey such a commutative law of multiplication, for example AB = BA for
A= 7 and B = 11, whereas in non-commutative multiplication, one would have AB 6= BA.
John Bell’s theorem asserts that local hidden variables plus the usual rules for probabilities
implycertaininequalitiesthatarenotsatisfiedbyquantummechanicalsystems–andexperiment
sees that these inequalities are in fact violated. There is much discussion of possible loopholes,
but I believe the result is robust, and that local hidden variables are excluded.
The other possibility is that the hidden variables are non-local: the hidden variables can act
faster than the speed of light to establish correlations (as long as no faster than light signaling
is possible). The hidden variables can also obey a non-commutative multiplication law.
Fortherestofthetalk I’llfocusonthepossibility ofnon-localhiddenvariables–thisis where
my research interests lie. At a phenomenological level, there are very interesting models for the
emergenceofprobabilitieswithintheusualwavefunctionformulationofnonrelativisticquantum
theory, pioneered by Ghirardi, Rimini, and Weber in Trieste and by also by Pearle at Hamilton
College in the U. S., and worked on by many others. These models postulate that space is
filled with a very low level noise with a coupling to matter proportional to the imaginary unit i,
ratherthanwithareal-valuedcoupling(moretechnically, theycouplethroughananti-Hermitian
Hamiltonian term). For example, there could be a small, rapidly fluctuating contribution to the
gravitational potential or g metric component proportional to the imaginary unit i. If such a
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theory obeys two general properties, (1) the total probability of a particle beingpresent remains
one for all times (that is, the wave function normalization is preserved), and (2) there is no
faster than light signaling, then the extra terms in the Schro¨dinger equation equation must have
a special structure. This special structure allows one to prove definite outcomes obeying the
Born rule!
In these models, for each repetition of the Stern–Gerlach experiment, the noise variable takes
different values. For a large apparatus, these have a measurable effect, whereas for an atom not
interactingwithanapparatus,theeffectisnotmeasurable. Thenoiseleadstodifferentoutcomes
for different runs, with probabilities given by the Born rule. The different noises for different
runs of the experiment are analogous, in the coin toss example I gave earlier, to different details
of the tumbling coin trajectories for the different coin tosses (Fig. 4).
I’ve worked on phenomenological reduction models, but my main long term interest has been
atthefoundationallevel. I amtryingto makeananalogy between quantummechanics emerging
froma possiblepre-quantum theory, andthe knownfact that thermodynamicsemerges from the
laws of statistical mechanics. Thermodynamics– the science of heat and work (Fig. 5) – reflects
PHENOMENOLOGICAL REDUCTION MODELS
UP
NOISE
RUN 1
RUN 2
DOWN
Figure 4. Different noisehistories, inobjective reductionmodels, canexplain “up”and“down”
registrations in the Stern–Gerlach experiment
averaged properties of huge numbers of atoms. It is a complete, consistent system by itself, and
remarkably was discovered in the 19th century before the existence of atoms was established.
But from statistical mechanics – the laws of motion of large systems of atoms, one can deduce
thelaws of thermodynamics,together withdetails offluctuation corrections to thermodynamics,
socalled Brownianmotioncorrections. (Thefigureshowstherandomwalk trajectory ofapollen
grain being bombarded by molecules in thermal motion.)
My suggestion, in articles with collaborators and a small book that I wrote in 2004 [1] , is a
theory that I call “trace dynamics”. It is a classical-like system of non-commutating variables –
evendistantsystemsintheuniverseareinteractinginstantaneouslywithus. Onecanmakesense
of the mathematics of non-commuting variables by using the cyclic property of a mathematical
operation called the Trace: TraceABCD..FG = TraceGABCD..F = TraceFGBCD... This is
a very powerful tool. One can use it to set up a system of equations analogous to classical
mechanics, and to do statistical averaging. Getting a little technical now, for those in the
audiencefamiliarwithquantumtheoryandstatisticalphysics,whatdistinguishestracedynamics
from ordinary classical mechanics is the existence of a generic conserved quantity in addition to
the energy and momentum. This quantity is operator-valued, and has the form
Conserved operator in trace dynamics =
P [q p −p q ]
bosons boson boson boson boson
− P [q p +p q ] ,
fermions fermion fermon fermion fermion
with the qs the canonical coordinates and the ps the canonical momenta. This is reminiscent
in structure to the canonical commutation and anti-commutation relations of quantum theory.
Just as energy in statistical mechanics is equally partitioned between the various degrees of
freedom, one might expect this conserved operator, in a statistical mechanical treatment of
trace dynamics, to also be equi-partitioned, giving the starting point for quantum theory.
My conjectures thus are: statistical averages in trace dynamics give the Schro¨dinger equation
(a)
THERMODYNAMICS
HEAT WORK
(b)
BROWNIAN MOTION
Figure 5. (a) Thermodynamics: the science of heat and work (b) Brownian motion of a pollen
grain being bombarded by molecules in the liquid in which it is suspended
and operator algebra of quantum theory, while Brownian motion corrections give the low level
noise on which phenomenological reduction models are based. I talked about this program
in my keynote address at the Vienna conference two years ago. Currently I am working on
incorporating gravity into trace dynamics [2] , and that is what I will talk about tomorrow. My
approach to an emergent quantum theory is still a work in progress – there is much yet to be
done!
Acknowledgments
IwishtothankGerhardGro¨ssingfororganizingtheconferenceEmQM13onEmergentQuantum
Mechanics, the Fetzer Franklin fund for its financial support, and Susan Higgins for redrawing
my figure sketches for publication.
References
[1] S.L.AdlerQuantum Theoryas anEmergent Phenomenon: TheStatistical Mechanics ofMatrix Modelsasthe
Precursor of Quantum Field Theory, Cambridge University Press, Cambridge (2004).
[2] S. L. Adler, “Incorporating gravity into trace dynamics: the induced gravitational action”, arXiv:1306.0482,
Class.QuantumGrav.30(2013)195015andClass.QuantumGrav.30(2013)239501(Erratum);S.L.Adler
andF.R.Ramazanoˇglu,“Sphericallysymmetricvacuumsolutionsarisingfromtracedynamicsmodifications
to gravitation”, arXiv:1308.1448.